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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.constants_faq"></a><a class="link" href="constants_faq.html" title="Math Constants FAQs">Math Constants FAQs</a>
</h2></div></div></div>
<h5>
<a name="math_toolkit.constants_faq.h0"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.why_are_these_constants_chosen"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_are_these_constants_chosen">Why are
      <span class="emphasis"><em>these</em></span> Constants Chosen?</a>
    </h5>
<p>
      It is, of course, impossible to please everyone with a list like this.
    </p>
<p>
      Some of the criteria we have used are:
    </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          Used in Boost.Math.
        </li>
<li class="listitem">
          Commonly used.
        </li>
<li class="listitem">
          Expensive to compute.
        </li>
<li class="listitem">
          Requested by users.
        </li>
<li class="listitem">
          <a href="http://en.wikipedia.org/wiki/Mathematical_constant" target="_top">Used in
          science and mathematics.</a>
        </li>
<li class="listitem">
          No integer values (because so cheap to construct).
        </li>
</ul></div>
<p>
      (You can easily define your own if found convenient, for example: <code class="computeroutput"><span class="identifier">FPT</span> <span class="identifier">one</span> <span class="special">=</span><span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">FPT</span><span class="special">&gt;(</span><span class="number">42</span><span class="special">);</span></code>).
    </p>
<h5>
<a name="math_toolkit.constants_faq.h1"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.how_are_constants_named"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_are_constants_named">How
      are constants named?</a>
    </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          Not macros, so no upper case.
        </li>
<li class="listitem">
          All lower case (following C++ standard names).
        </li>
<li class="listitem">
          No CamelCase.
        </li>
<li class="listitem">
          Underscore as _ delimiter between words.
        </li>
<li class="listitem">
          Numbers spelt as words rather than decimal digits (except following pow).
        </li>
<li class="listitem">
          Abbreviation conventions:
          <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
<li class="listitem">
                root for square root.
              </li>
<li class="listitem">
                cbrt for cube root.
              </li>
<li class="listitem">
                pow for pow function using decimal digits like pow23 for n<sup>2/3</sup>.
              </li>
<li class="listitem">
                div for divided by or operator /.
              </li>
<li class="listitem">
                minus for operator -, plus for operator +.
              </li>
<li class="listitem">
                sqr for squared.
              </li>
<li class="listitem">
                cubed for cubed n<sup>3</sup>.
              </li>
<li class="listitem">
                words for greek, like π, ζ and Γ.
              </li>
<li class="listitem">
                words like half, third, three_quarters, sixth for fractions. (Digit(s)
                can get muddled).
              </li>
<li class="listitem">
                log10 for log<sub>10</sub>
              </li>
<li class="listitem">
                ln for log<sub>e</sub>
              </li>
</ul></div>
        </li>
</ul></div>
<h5>
<a name="math_toolkit.constants_faq.h2"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.how_are_the_constants_derived"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_are_the_constants_derived">How are
      the constants derived?</a>
    </h5>
<p>
      The constants have all been calculated using high-precision software working
      with up to 300-bit precision giving about 100 decimal digits. (The precision
      can be arbitrarily chosen and is limited only by compute time).
    </p>
<h5>
<a name="math_toolkit.constants_faq.h3"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.how_accurate_are_the_constants"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_accurate_are_the_constants">How Accurate
      are the constants?</a>
    </h5>
<p>
      The minimum accuracy chosen (100 decimal digits) exceeds the accuracy of reasonably-foreseeable
      floating-point hardware (256-bit) and should meet most high-precision computations.
    </p>
<h5>
<a name="math_toolkit.constants_faq.h4"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.how_are_the_constants_tested"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_are_the_constants_tested">How are the
      constants tested?</a>
    </h5>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
          Comparison using Boost.Test BOOST_CHECK_CLOSE_FRACTION using long double
          literals, with at least 35 decimal digits, enough to be accurate for all
          long double implementations. The tolerance is usually twice <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">epsilon</span></code>.
        </li>
<li class="listitem">
          Comparison with calculation at long double precision. This often requires
          a slightly higher tolerance than two epsilon because of computational noise
          from round-off etc, especially when trig and other functions are called.
        </li>
<li class="listitem">
          Comparison with independent published values, for example, using <a href="http://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences (OEIS)</a>
          again using at least 35 decimal digits strings.
        </li>
<li class="listitem">
          Comparison with independently calculated values using arbitrary precision
          tools like <a href="http://www.wolfram.com/mathematica/" target="_top">Mathematica</a>,
          again using at least 35 decimal digits literal strings.
        </li>
</ol></div>
<div class="warning"><table border="0" summary="Warning">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../doc/src/images/warning.png"></td>
<th align="left">Warning</th>
</tr>
<tr><td align="left" valign="top"><p>
        We have not yet been able to <span class="bold"><strong>check</strong></span> that
        <span class="bold"><strong>all</strong></span> constants are accurate at the full arbitrary
        precision, at present 100 decimal digits. But certain key values like <code class="computeroutput"><span class="identifier">e</span></code> and <code class="computeroutput"><span class="identifier">pi</span></code>
        appear to be accurate and internal consistencies suggest that others are
        this accurate too.
      </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.constants_faq.h5"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.why_is_portability_important"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_is_portability_important">Why is Portability
      important?</a>
    </h5>
<p>
      Code written using math constants is easily portable even when using different
      floating-point types with differing precision.
    </p>
<p>
      It is a mistake to expect that results of computations will be <span class="bold"><strong>identical</strong></span>,
      but you can achieve the <span class="bold"><strong>best accuracy possible for the
      floating-point type in use</strong></span>.
    </p>
<p>
      This has no extra cost to the user, but reduces irritating, and often confusing
      and very hard-to-trace effects, caused by the intrinsically limited precision
      of floating-point calculations.
    </p>
<p>
      A harmless symptom of this limit is a spurious least-significant digit; at
      worst, slightly inaccurate constants sometimes cause iterating algorithms to
      diverge wildly because internal comparisons just fail.
    </p>
<h5>
<a name="math_toolkit.constants_faq.h6"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.what_is_the_internal_format_of_t"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_is_the_internal_format_of_t">What
      is the Internal Format of the constants, and why?</a>
    </h5>
<p>
      See <a class="link" href="tutorial.html" title="Tutorial">tutorial</a> above for normal
      use, but this FAQ explains the internal details used for the constants.
    </p>
<p>
      Constants are stored as 100 decimal digit values. However, some compilers do
      not accept decimal digits strings as long as this. So the constant is split
      into two parts, with the first containing at least 128-bit long double precision
      (35 decimal digits), and for consistency should be in scientific format with
      a signed exponent.
    </p>
<p>
      The second part is the value of the constant expressed as a string literal,
      accurate to at least 100 decimal digits (in practice that means at least 102
      digits). Again for consistency use scientific format with a signed exponent.
    </p>
<p>
      For types with precision greater than a long double, then if T is constructible
      <code class="computeroutput"><span class="identifier">T</span> </code>is constructible from a
      <code class="computeroutput"><span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span></code> then it's directly constructed from the string,
      otherwise we fall back on lexical_cast to convert to type <code class="computeroutput"><span class="identifier">T</span></code>.
      (Using a string is necessary because you can't use a numeric constant since
      even a <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
      might not have enough digits).
    </p>
<p>
      So, for example, a constant like pi is internally defined as
    </p>
<pre class="programlisting"><span class="identifier">BOOST_DEFINE_MATH_CONSTANT</span><span class="special">(</span><span class="identifier">pi</span><span class="special">,</span> <span class="number">3.141592653589793238462643383279502884e+00</span><span class="special">,</span> <span class="string">"3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651e+00"</span><span class="special">);</span>
</pre>
<p>
      In this case the significand is 109 decimal digits, ensuring 100 decimal digits
      are exact, and exponent is zero.
    </p>
<p>
      See <a class="link" href="new_const.html" title="Defining New Constants">defining new constants</a> to
      calculate new constants.
    </p>
<p>
      A macro definition like this can be pasted into user code where convenient,
      or into <code class="computeroutput"><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">constants</span><span class="special">.</span><span class="identifier">hpp</span></code> if it
      is to be added to the Boost.Math library.
    </p>
<h5>
<a name="math_toolkit.constants_faq.h7"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.what_floating_point_types_could_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_floating_point_types_could_">What
      Floating-point Types could I use?</a>
    </h5>
<p>
      Apart from the built-in floating-point types <code class="computeroutput"><span class="keyword">float</span></code>,
      <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
      <span class="keyword">double</span></code>, there are several arbitrary
      precision floating-point classes available, but most are not licensed for commercial
      use.
    </p>
<h6>
<a name="math_toolkit.constants_faq.h8"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.boost_multiprecision_by_christop"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.boost_multiprecision_by_christop">Boost.Multiprecision
      by Christopher Kormanyos and John Maddock</a>
    </h6>
<p>
      This is a well-maintained and fully featured library which provides numerous
      high-precision types, as well as wrappers to other popular libraries like MPFR.
    </p>
<h6>
<a name="math_toolkit.constants_faq.h9"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.ntl_class_quad_float"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.ntl_class_quad_float">NTL
      class quad_float</a>
    </h6>
<p>
      <a href="http://shoup.net/ntl/" target="_top">NTL</a> by Victor Shoup has fixed and
      arbitrary high precision fixed and floating-point types. However none of these
      are licenced for commercial use.
    </p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">NTL</span><span class="special">/</span><span class="identifier">quad_float</span><span class="special">.</span><span class="identifier">h</span><span class="special">&gt;</span> <span class="comment">// quad precision 106-bit, about 32 decimal digits.</span>
<span class="keyword">using</span> <span class="identifier">NTL</span><span class="special">::</span><span class="identifier">to_quad_float</span><span class="special">;</span> <span class="comment">// Less precise than arbitrary precision NTL::RR.</span>
</pre>
<p>
      NTL class <code class="computeroutput"><span class="identifier">quad_float</span></code>, which
      gives a form of quadruple precision, 106-bit significand (but without an extended
      exponent range.) With an IEC559/IEEE 754 compatible processor, for example
      Intel X86 family, with 64-bit double, and 53-bit significand, using the significands
      of <span class="bold"><strong>two</strong></span> 64-bit doubles, if <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">digits10</span></code> is 16, then we get about twice the
      precision, so <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">quad_float</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">()</span></code>
      should be 32. (the default <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">RR</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">()</span></code>
      should be about 40). (which seems to agree with experiments). We output constants
      (including some noisy bits, an approximation to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">RR</span><span class="special">&gt;::</span><span class="identifier">max_digits10</span><span class="special">()</span></code>)
      by adding 2 or 3 extra decimal digits, so using <code class="computeroutput"><span class="identifier">quad_float</span><span class="special">::</span><span class="identifier">SetOutputPrecision</span><span class="special">(</span><span class="number">32</span> <span class="special">+</span>
      <span class="number">3</span><span class="special">);</span></code>
    </p>
<p>
      Apple Mac/Darwin uses a similar <span class="emphasis"><em>doubledouble</em></span> 106-bit for
      its built-in <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
      type.
    </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
        The precision of all <code class="computeroutput"><span class="identifier">doubledouble</span></code>
        floating-point types is rather odd and values given are only approximate.
      </p></td></tr>
</table></div>
<p>
      <span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
    </p>
<h6>
<a name="math_toolkit.constants_faq.h10"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.ntl_class_rr"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.ntl_class_rr">NTL
      class RR</a>
    </h6>
<p>
      Arbitrary precision floating point with NTL class RR, default is 150 bit (about
      50 decimal digits) used here with 300 bit to output 100 decimal digits, enough
      for many practical non-'number-theoretic' C++ applications.
    </p>
<p>
      <a href="http://www.shoup.net/ntl/" target="_top">NTL A Library for doing Number Theory</a>
      is <span class="bold"><strong>not licenced for commercial use</strong></span>.
    </p>
<p>
      This class is used in Boost.Math and is an option when using big_number projects
      to calculate new math constants.
    </p>
<p>
      <span class="bold"><strong>New projects should use <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span>
    </p>
<h6>
<a name="math_toolkit.constants_faq.h11"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.gmp_and_mpfr"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.gmp_and_mpfr">GMP
      and MPFR</a>
    </h6>
<p>
      <a href="http://gmplib.org" target="_top">GMP</a> and <a href="http://www.mpfr.org/" target="_top">MPFR</a>
      have also been used to compute constants, but are licensed under the <a href="http://www.gnu.org/copyleft/lesser.html" target="_top">Lesser GPL license</a> and
      are <span class="bold"><strong>not licensed for commercial use</strong></span>.
    </p>
<h5>
<a name="math_toolkit.constants_faq.h12"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.what_happened_to_a_previous_coll"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_happened_to_a_previous_coll">What
      happened to a previous collection of constants proposed for Boost?</a>
    </h5>
<p>
      A review concluded that the way in which the constants were presented did not
      meet many peoples needs. None of the methods proposed met many users' essential
      requirement to allow writing simply <code class="computeroutput"><span class="identifier">pi</span></code>
      rather than <code class="computeroutput"><span class="identifier">pi</span><span class="special">()</span></code>.
      Many science and engineering equations look difficult to read because function
      call brackets can be confused with the many other brackets often needed. All
      the methods then proposed of avoiding the brackets failed to meet all needs,
      often on grounds of complexity and lack of applicability to various realistic
      scenarios.
    </p>
<p>
      So the simple namespace method, proposed on its own, but rejected at the first
      review, has been added to allow users to have convenient access to <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>
      and <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
      values, but combined with template struct and functions to allow simultaneous
      use with other non-built-in floating-point types.
    </p>
<h5>
<a name="math_toolkit.constants_faq.h13"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.why_do_the_constants_internally_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_do_the_constants_internally_">Why do
      the constants (internally) have a struct rather than a simple function?</a>
    </h5>
<p>
      A function mechanism was provided by in previous versions of Boost.Math.
    </p>
<p>
      The new mechanism is to permit partial specialization. See Custom Specializing
      a constant above. It should also allow use with other packages like <a href="http://www.ttmath.org/" target="_top">ttmath Bignum C++ library.</a>
    </p>
<h5>
<a name="math_toolkit.constants_faq.h14"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.where_can_i_find_other_high_prec"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.where_can_i_find_other_high_prec">Where
      can I find other high precision constants?</a>
    </h5>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
          Constants with very high precision and good accuracy (&gt;40 decimal digits)
          from Simon Plouffe's web based collection <a href="http://pi.lacim.uqam.ca/eng/" target="_top">http://pi.lacim.uqam.ca/eng/</a>.
        </li>
<li class="listitem">
          <a href="https://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences
          (OEIS)</a>
        </li>
<li class="listitem">
          Checks using printed text optically scanned values and converted from:
          D. E. Knuth, Art of Computer Programming, Appendix A, Table 1, Vol 1, ISBN
          0 201 89683 4 (1997)
        </li>
<li class="listitem">
          M. Abrahamovitz &amp; I. E. Stegun, National Bureau of Standards, Handbook
          of Mathematical Functions, a reference source for formulae now superseded
          by
        </li>
<li class="listitem">
          Frank W. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark,
          NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN
          978-0-521-14063-8, 2010.
        </li>
<li class="listitem">
          John F Hart, Computer Approximations, Kreiger (1978) ISBN 0 88275 642 7.
        </li>
<li class="listitem">
          Some values from Cephes Mathematical Library, Stephen L. Moshier and CALC100
          100 decimal digit Complex Variable Calculator Program, a DOS utility.
        </li>
<li class="listitem">
          Xavier Gourdon, Pascal Sebah, 50 decimal digits constants at <a href="http://numbers.computation.free.fr/Constants/constants.html" target="_top">Number,
          constants and computation</a>.
        </li>
</ol></div>
<h5>
<a name="math_toolkit.constants_faq.h15"></a>
      <span class="phrase"><a name="math_toolkit.constants_faq.where_are_physical_constants"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.where_are_physical_constants">Where are
      Physical Constants?</a>
    </h5>
<p>
      Not here, because physical constants:
    </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          Are measurements, not truly constants.
        </li>
<li class="listitem">
          Are not truly constant and keeping changing as metrology improves.
        </li>
<li class="listitem">
          Have intrinsic uncertainty.
        </li>
<li class="listitem">
          Mathematical constants are stored and represented at varying precision,
          but should never be inaccurate.
        </li>
</ul></div>
<p>
      Some physical constants may be available in Boost.Units.
    </p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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